This paper is devoted to generalizing the trace-squared (conjugation) invariant of Möbius transformations with complex entries to elements of a certain set of Clifford matrices . These were first considered by K. Th. Vahlen [Math. Ann. 55, 585-593 (1902)] and later by L. V. Ahlfors [see e.g. Ann. Acad. Sci. Fenn., Ser. A I 105, 15-27 (1985; Zbl 0586.30045)]. The Clifford matrices considered here are slightly different from those obtained by Ahlfors and provide a generalization to orientation-reversing maps. The main theorem is that, for each , the function (the definition is too complicated to give here)
is a conjugation invariant. Here is the algebra of Clifford numbers generated by n roots of -1. The conjugation is by Clifford matrices. Among the properties of the invariant are the following:
(i) for all . Thus is well-defined on the projectivization of
(ii) for all k odd (resp. even) if g is orientation preserving (resp. reversing)
(iii) is hyperbolic if and only if for the largest k so that
(iv) is elliptic if and only if the quadratic form , when restricted to the kernel of Id-r(g) is not positive semidefinite. Here r takes g from the isometries of the ball model of hyperbolic space to the hyperboloid model.